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Properties

Label 392.3.k.h
Level $392$
Weight $3$
Character orbit 392.k
Analytic conductor $10.681$
Analytic rank $0$
Dimension $4$
CM discriminant -8
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [392,3,Mod(67,392)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(392, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 3, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("392.67");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 392 = 2^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 392.k (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.6812263629\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - 4 \beta_{2} - 4) q^{4} - 2 \beta_{3} q^{6} - 8 q^{8} + 23 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_{2} q^{2} + \beta_1 q^{3} + ( - 4 \beta_{2} - 4) q^{4} - 2 \beta_{3} q^{6} - 8 q^{8} + 23 \beta_{2} q^{9} + (14 \beta_{2} + 14) q^{11} + ( - 4 \beta_{3} - 4 \beta_1) q^{12} + 16 \beta_{2} q^{16} + 6 \beta_1 q^{17} + (46 \beta_{2} + 46) q^{18} + ( - 3 \beta_{3} - 3 \beta_1) q^{19} + 28 q^{22} - 8 \beta_1 q^{24} + ( - 25 \beta_{2} - 25) q^{25} + 14 \beta_{3} q^{27} + (32 \beta_{2} + 32) q^{32} + (14 \beta_{3} + 14 \beta_1) q^{33} - 12 \beta_{3} q^{34} + 92 q^{36} - 6 \beta_1 q^{38} - 12 \beta_{3} q^{41} - 14 q^{43} - 56 \beta_{2} q^{44} + 16 \beta_{3} q^{48} - 50 q^{50} + 192 \beta_{2} q^{51} + (28 \beta_{3} + 28 \beta_1) q^{54} + 96 q^{57} - 15 \beta_1 q^{59} + 64 q^{64} + 28 \beta_1 q^{66} + ( - 62 \beta_{2} - 62) q^{67} + ( - 24 \beta_{3} - 24 \beta_1) q^{68} - 184 \beta_{2} q^{72} - 6 \beta_1 q^{73} + ( - 25 \beta_{3} - 25 \beta_1) q^{75} + 12 \beta_{3} q^{76} + ( - 241 \beta_{2} - 241) q^{81} + ( - 24 \beta_{3} - 24 \beta_1) q^{82} - 9 \beta_{3} q^{83} + 28 \beta_{2} q^{86} + ( - 112 \beta_{2} - 112) q^{88} + ( - 18 \beta_{3} - 18 \beta_1) q^{89} + (32 \beta_{3} + 32 \beta_1) q^{96} - 30 \beta_{3} q^{97} - 322 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 8 q^{4} - 32 q^{8} - 46 q^{9} + 28 q^{11} - 32 q^{16} + 92 q^{18} + 112 q^{22} - 50 q^{25} + 64 q^{32} + 368 q^{36} - 56 q^{43} + 112 q^{44} - 200 q^{50} - 384 q^{51} + 384 q^{57} + 256 q^{64} - 124 q^{67} + 368 q^{72} - 482 q^{81} - 56 q^{86} - 224 q^{88} - 1288 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 2x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 2\nu^{3} \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/392\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(297\)
\(\chi(n)\) \(-1\) \(-1\) \(\beta_{2}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
67.1
−0.707107 + 1.22474i
0.707107 − 1.22474i
−0.707107 − 1.22474i
0.707107 + 1.22474i
1.00000 + 1.73205i −2.82843 + 4.89898i −2.00000 + 3.46410i 0 −11.3137 0 −8.00000 −11.5000 − 19.9186i 0
67.2 1.00000 + 1.73205i 2.82843 − 4.89898i −2.00000 + 3.46410i 0 11.3137 0 −8.00000 −11.5000 − 19.9186i 0
275.1 1.00000 − 1.73205i −2.82843 − 4.89898i −2.00000 − 3.46410i 0 −11.3137 0 −8.00000 −11.5000 + 19.9186i 0
275.2 1.00000 − 1.73205i 2.82843 + 4.89898i −2.00000 − 3.46410i 0 11.3137 0 −8.00000 −11.5000 + 19.9186i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner
56.e even 2 1 inner
56.k odd 6 1 inner
56.m even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 392.3.k.h 4
7.b odd 2 1 inner 392.3.k.h 4
7.c even 3 1 392.3.g.b ✓ 2
7.c even 3 1 inner 392.3.k.h 4
7.d odd 6 1 392.3.g.b ✓ 2
7.d odd 6 1 inner 392.3.k.h 4
8.d odd 2 1 CM 392.3.k.h 4
28.f even 6 1 1568.3.g.e 2
28.g odd 6 1 1568.3.g.e 2
56.e even 2 1 inner 392.3.k.h 4
56.j odd 6 1 1568.3.g.e 2
56.k odd 6 1 392.3.g.b ✓ 2
56.k odd 6 1 inner 392.3.k.h 4
56.m even 6 1 392.3.g.b ✓ 2
56.m even 6 1 inner 392.3.k.h 4
56.p even 6 1 1568.3.g.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
392.3.g.b ✓ 2 7.c even 3 1
392.3.g.b ✓ 2 7.d odd 6 1
392.3.g.b ✓ 2 56.k odd 6 1
392.3.g.b ✓ 2 56.m even 6 1
392.3.k.h 4 1.a even 1 1 trivial
392.3.k.h 4 7.b odd 2 1 inner
392.3.k.h 4 7.c even 3 1 inner
392.3.k.h 4 7.d odd 6 1 inner
392.3.k.h 4 8.d odd 2 1 CM
392.3.k.h 4 56.e even 2 1 inner
392.3.k.h 4 56.k odd 6 1 inner
392.3.k.h 4 56.m even 6 1 inner
1568.3.g.e 2 28.f even 6 1
1568.3.g.e 2 28.g odd 6 1
1568.3.g.e 2 56.j odd 6 1
1568.3.g.e 2 56.p even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(392, [\chi])\):

\( T_{3}^{4} + 32T_{3}^{2} + 1024 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 32T^{2} + 1024 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} + 1152 T^{2} + 1327104 \) Copy content Toggle raw display
$19$ \( T^{4} + 288 T^{2} + 82944 \) Copy content Toggle raw display
$23$ \( T^{4} \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 4608)^{2} \) Copy content Toggle raw display
$43$ \( (T + 14)^{4} \) Copy content Toggle raw display
$47$ \( T^{4} \) Copy content Toggle raw display
$53$ \( T^{4} \) Copy content Toggle raw display
$59$ \( T^{4} + 7200 T^{2} + 51840000 \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( (T^{2} + 62 T + 3844)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 1152 T^{2} + 1327104 \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} - 2592)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + 10368 T^{2} + 107495424 \) Copy content Toggle raw display
$97$ \( (T^{2} - 28800)^{2} \) Copy content Toggle raw display
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